Why do bicycles have two individual brake levers that work independently to brake each wheel? I've always used my right brake lever because it is more comfortable to do so and I haven't noticed any difference between the two.

When reading people's answers, note that whether the left hand or the right hand controls the front brake or the back brake may depend on which country you're in (or, more specifically, which country your bike is designed for).
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ChrisWMay 27 '11 at 23:35

@ChrisW - A very good point. However, since this isn't answer on its own, will make this a comment. @Ambo100 - right brake lever = front or rear wheel on your bike?
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Neil FeinMay 28 '11 at 13:29

Allows signaling with either hand and stopping at the same time. Sometimes you need to be slowing down to prepare for a stop or turn and signaling at the same time. If you can take one hand or the other (but not both) off the handlebars that can be useful. (I prefer the arm straight out in direction of turn signal instead of the left arm hooked up to signal right turn)

Simplicity of design. A single lever controlling both brakes (via two cables?) would be tricky to adjust right, and if it wasn't adjusted exactly perfectly you'd probably end up braking with one wheel and the other never really happening. The longer cable (rear brake) would get more stretch (housing compression) just because it's longer, so you'd probably need to be fiddling with barrel adjusters on that cable quite frequently. One lever controlling both brakes just seems difficult to engineer right and likely to need much more frequent adjusting/maintenance.

Personally, I tend to use the rear brake for controlling my speed (slowing down just a little) and either the front brake or both brakes for stopping (especially a fast stop). An awareness at any given time of which tire is more likely to slip and using the other brake is good, though.

Worth noting: it's a specialty item, but you can get a brake lever that controls both brake levers at once. This is mostly popular with bike polo players (who use one hand for a mallet) and people with physical conditions that limit their ability to use the brake with one hand. If you don't have any special reasons to need just one lever, you're probably best off with separate levers. http://www.paulcomp.com/duplexlever.html

Linking the two brakes would have a detrimental effect on braking power.

Your front brake will bring your bike to a halt far quicker than your rear brake will, and should be used almost exclusively. When braking with the rear brake, your back tire won't have much weight on it, and will skid along the ground. This results in a dramatic reduction of braking power. On the other hand, when you compress your front brake, the forward momentum of the bike will be partially converted to downwards force, sticking the tire hard against the road surface and maintaining maximum traction. In a real-life scenario, this can easily be the difference between slamming into a car that suddenly brakes in front of you versus screeching to a halt with an inch to spare.

Of course, if you squeeze the front brake too hard at too great a speed, you can flip over the handlebars. However, if you primarily use your front brake, you will quickly learn exactly how quickly you can safely stop the bicycle. Being low in the drops helps with this immensely.

The few scenarios in which you should be using your rear brake are during fast downhill descents, where even a little braking on the front at fast speeds can cause your center of gravity to go over the front of the bike, or on any surface where your front wheel might lose contact with the ground. In this situation, the front brake could bring your front wheel to a complete stop just before it connects with the ground again, causing you to lose control.

I'd like to add that no matter what brake you use it's all about your posture when braking, if you lean back into the seat a bit more and put pressure on the back wheel you'll be less likely to flip.
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Ambo100May 26 '11 at 18:26

I will add that there can (theory) be one break "mechanisms" to activate two breaks. Look at a cars shoe break, which result in completely different pressure front/rear. Also even a little delay of the rear wheel-pair. The reason of that delay are very nice explained in your answer.
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IndependentMay 27 '11 at 4:51

It's about security that (at least) two completely independent break mechanisms should exist. If one mechanism fails, the other are there to serve. In competitions and other special circumstances like high speed, slippery roads and down hills, you also are helped with choose front / rear depends on the situation.

Actually, this is a law in many places. You have to have 2 completely independent breaking systems. Which is why children's bikes will often have a regular front brake along with the coaster brake. Even though my kids (5 and 3) don't have the hand strength to use the front hand brake. Also fixed gear bikes are required to have at least 1 hand brake in some places for the same reason. The fixed gear configuration counts, but you still have to have a backup. If the chain jumps while you're going downhill, you'll be hoping you have handbrakes on that fixie.
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KibbeeJun 2 '11 at 18:49

I note that no one has mentioned that you want to have separate control over front and rear to prevent flipping. [Well, on re-reading, I see that Stephen did.] The front brake carries about 80% of the braking power, but if it locks up the bike flips rather quickly. If the rear brake locks up the bike will tend to slide sideways. When attempting to stop as rapidly as possible one should apply the most force (2-3x) to the front brake but then, if the rear wheel begins to slip, release the FRONT brake. (The rear wheel will slip because it's getting "lighter" due to the front brake being near lock-up.)

It's fairly simple; to stop effectively, you need to apply the two controls in a different manner.

When you start braking, you have a given amount of weight on the front and rear wheels. As you apply the brakes, the deceleration shifts weight from the rear wheels to the front wheels, which gives more traction at the front, and less traction at the rear. More traction at the front means that you can apply the brakes more there, less traction at the rear means that you need to reduce the power on the rear brake.

When I used to teach motorcycle safety, we would say "progressive squeeze on the front brake, light to lighter pressure on the rear".